Question

If the LCM of the polynomials $f\left(x\right)=(x+1{)}^5(x+2{)}^a$ and $g\left(x\right)=(x+1{)}^b(x+2{)}^a$ is $(x+1{)}^a(x+2{)}^b$, then find the minimum value of $a+b$ from the following options.

• A. 10
• B. 14
• C. 15
• D. Cannot say
  

Answer 1

The correct option is A 10

$f\left(x\right)=(x+1{)}^5(x+2{)}^a$

$g\left(x\right)=(x+1{)}^b(x+2{)}^5$

LCM = $(x+1{)}^a(x+2{)}^b$

As $(x+1{)}^a$ is a multiple of both $(x+1{)}^5$ and $(x+1{)}^b,a\ge 5$ and $a\ge b$ .....(1)

As $(x+2{)}^b$ is a multiple of both $(x+2{)}^a$ and $(x+2{)}^5,b\ge 5$ and $b\ge a$ ...........(2)

From equation 1 & 2,

$a=b,a\ge 5$ and $b\ge 5$

Therefore, minimum value of $a+b$ is $5+5,i.e.,10.$